At last, Michael Clarke must have thought. At last Australia have tasted victory this summer. It took until the fourth Test of their home campaign, and into the final hour at that, but Mitchell Starc and Peter Siddle delivered a 137-run win for Clarke and their team-mates as Sri Lanka's tail-enders narrowly failed to hold on for a draw at Bellerive Oval. Despite the best efforts of Kumar Sangakkara and Thilan Samaraweera in particular, Sri Lanka will walk out on Boxing Day at the MCG 1-0 down.

As the gloom began to close in, Australia entered the final hour needing two more wickets; Sri Lanka's victory target of 393 had become irrelevant during the morning. Starc delivered precisely what his captain required. All the Twenty20 cricket he has played over the past year began to pay off. He viewed the task as similar to bowling at the death in the short format, sending in yorkers and mixing it up with the occasional shorter delivery. And unlike in T20, he had the luxury of catching men everywhere.

After a few yorkers narrowly missed the stumps, or hit the pads on their way down the leg side, Starc finally directed one that bowled Rangana Herath to leave Sri Lanka at 9 for 250. In his next over, Starc banged one in shorter and Shaminda Eranga couldn't work out how to play it, in the end offering a fend of sorts that was edged through to Matthew Wade. Starc had 5 for 63. Sri Lanka were out for 255. And Australia had won the fourth Test of their home season, the first time since 2001-02 it had taken them so long to register a win.

There was a sense of inevitability about the victory, at least, after Samaraweera and Sangakkara had departed. Unlike in the Adelaide Test last month, where the South Africans held off Australia's bowlers to survive for a draw, this time the pitch was tougher for batting, the weather was much cooler and allowed the bowlers some respite, and importantly, it wasn't all left to Siddle. It was him who provided the spark, though.

Siddle was named Man of the Match for his nine-wicket game, and most importantly for the Australians he provided the three key breakthroughs on the final day. First there was Mahela Jayawardene, who before lunch edged a ball that moved away from the bat and was caught at slip for 19. Then it was the big one - Sangakkara lbw for 63. It was a perfect delivery, angling across the left-hander, pitching in line and straightening. A hopeful review did not save the batsman.

Australia were pleased to have a review go their way after they had both Sangakkara and Samaraweera given out lbw earlier in the day only to have the decisions overturned. On 54, Sangakkara had been reprieved when he tried to pull a Shane Watson delivery that kept low and after being given out lbw, he asked for a review of Nigel Llong's decision. The replays showed Watson, who was coming around the wicket, had struck Sangakkara just outside the line of off stump.

Samaraweera had been on 18 when he was given lbw to Siddle, but replays showed Llong had erred again, for the ball had clearly struck the batsman outside the line of off stump while playing a shot. Samaraweera provided plenty of fight after that, surviving until after tea, when Siddle jagged a ball back to have him lbw for 49. Siddle had already had Angelo Mathews caught behind for 19, and Australia were into the tail.

Smart stats

Peter Siddle's match haul of 9 for 104 is his best in Tests surpassing his previous best of 8 for 113 in Sydney in 2009.

Siddle's match performance is the third-best for an Australian bowler against Sri Lanka. Shane Warne is the only Australian bowler to pick up ten-wicket match hauls (twice) against Sri Lanka.

Siddle's 9 for 104 is the third-best bowling display in a Test in Hobart. Doug Bracewell (9 for 60) and Warne (9 for 67) are ahead of Siddle.

Mitchell Starc's 5 for 63 is his second-best bowling performance after the six-wicket haul against South Africa in Perth earlier this year.

The 137-run win is Australia's third in three Tests against Sri Lanka in Hobart. Overall, Australia have won nine of the 11 home Tests against Sri Lanka.

The number of overs Sri Lanka batted in the fourth innings (119.2) is the third-highest for them in Tests outside the subcontinent. The highest (141.4) also came in Hobart in 1989 when they lost by 173 runs.

Then it was all Starc. Supported by a predatory field that featured every player in the normal television frame - on several occasions Australia had no fielder who could not have been called a catcher - Starc picked up the key wicket of Prasanna Jayawardene, the last of the recognised batsmen. Bowling around the wicket, Starc forced Jayawardene to play a ball that bounced more than he expected, and he was taken at slip for 21.

Starc then went back over the wicket and angled a ball across Nuwan Kulasekara, who edged behind for 9, and Australia could sniff victory. It was a fine effort from Starc and Siddle after Sri Lanka had been only three wickets down at lunch and four wickets down at tea.

Sangakkara had spent most of the first half of the day frustrating the Australians just as he had done at the same ground five years ago. He occupied the crease for 226 balls in a cautious innings that had included just six boundaries. He brought up his half-century from his 165th ball with a pull to the midwicket boundary off a generous full toss from David Warner.

Samaraweera was also watchful, although he showed a willingness to mix things up when he advanced to Nathan Lyon and lofted a boundary to long-on. He was the main problem for the Australians after Sangakkara departed, and on a day when rain had also frustrated Australia, Clarke was so desperate for a breakthrough that he asked the wicketkeeper Wade to bowl the final over before tea.

It was a remarkable move given that Wade had never before bowled a delivery in his 60-match first-class career. His right-arm medium pacers came through sharper than expected - a couple of balls clocked 132kph - but he was unable to force a mistake from Samaraweera.

Fortunately for Australia, the strike bowlers did the job in the final session. Sri Lanka's fight was admirable, but on a wearing pitch their task was simply too great.

Brydon Coverdale is an assistant editor at ESPNcricinfo. He tweets here